Pizzetti formulae for Stiefel manifolds and applications

TL;DR

This paper generalizes Pizzetti's formula to Stiefel manifolds, providing a new method for calculating group integrals and revealing algebraic structures relevant to random matrix theory applications.

Contribution

The authors extend Pizzetti's formula to real, complex, and quaternion Stiefel manifolds, unifying integration techniques and exploring algebraic structures like the Harish-Chandra integral.

Findings

Derived a unified formula for integrals over Stiefel manifolds.

Applied the formula to compute the Itzykson-Zuber integral for SO(4)/[SO(2)xSO(2)].

Connected the results to two-point correlation functions in random matrix theory.

Abstract

Pizzetti's formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson-Zuber integral for the coset SO(4)/[SO(2)xSO(2)]. This integral naturally appears in the calculation of the two-point correlation function in the transition of the statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in random matrix theory.